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View Article OnlineView Journal | View Issue

A model for grain

aFederal Institute for Materials Research a

12205 Berlin, Germany. E-mail: reza.kam

combMax-Planck-Institut fur Eisenforschung,

Germany

Cite this: RSC Adv., 2020, 10, 26728

Received 27th May 2020Accepted 29th June 2020

DOI: 10.1039/d0ra04682e

rsc.li/rsc-advances

26728 | RSC Adv., 2020, 10, 26728–

boundary thermodynamics

Reza Darvishi Kamachali *ab

Systematic microstructure design requires reliable thermodynamic descriptions of each and all

microstructure elements. While such descriptions are well established for most bulk phases,

thermodynamic assessment of microstructure defects is challenging because of their individualistic

nature. In this paper, a model is devised for assessing grain boundary thermodynamics based on available

bulk thermodynamic data. We propose a continuous relative atomic density field and its spatial gradients

to describe the grain boundary region with reference to the homogeneous bulk and derive the grain

boundary Gibbs free energy functional. The grain boundary segregation isotherm and phase diagram are

computed for a regular binary solid solution, and qualitatively benchmarked for the Pt–Au system. The

relationships between the grain boundary's atomic density, excess free volume, and misorientation angle

are discussed. Combining the current density-based model with available bulk thermodynamic databases

enables constructing databases, phase diagrams, and segregation isotherms for grain boundaries,

opening possibilities for studying and designing heterogeneous microstructures.

1. Introduction

Grain boundaries (GBs) are one of the main sources of hetero-geneity in polycrystalline microstructures; they interact withvarious microstructure elements, e.g. GB networks,1 secondary-phase particles,2–4 dislocations,5–7 and vacancies,8,9 while theythemselves possess distinct physical properties similar to thebulk phases. A critical source of microstructure heterogeneity isGBs' interaction with solute atoms that inuences both the localand global chemistry of the microstructure. Solute segregationto GBs canmediate a whole different sort of phenomena such assegregation-assisted GB premelting,10 phase transition,11–13

precipitation14,15 and embrittlement16,17 as well as stabilizationof nanocrystalline materials.18–20 Hence, understanding andcontrolling GBs and their spatial interactions with othermicrostructure elements can unravel new ideas for designingheterogeneous microstructures.

The studies on interfacial segregation can be traced back tothe works of Gibbs,21 Langmuir,22 McLean23 and Fowler andGuggenheim24 on the adsorption. Since then, numerous studieson the segregation phenomena have been conducted.25–42 vander Waals,43 Cahn and Hilliard44 and Cahn45 worked out freeenergy functionals that account for heterogeneous interfacialfeatures on the mesoscale. Based on this idea, several modelsfor GB segregation were proposed. Ma et al.46 developed a phase-eld model for studying GB segregation in which GB region was

nd Testing (BAM), Unter den Eichen 87,

achali@bam.de; reza.kamachali@gmail.

Max-Planck-Str. 1, 40237 Dusseldorf,

26741

described by its reduced atomic coordination number. Based onKobayashi–Warren–Carter's work,47 Tang et al. developedmodels for order-disorder transition48,49 and phase transition50

at GBs. Heo et al.51 proposed a model for GB segregation anddrag including mist elastic interactions. The common featureof these models is the non-vanishing gradients in GB structureand/or composition. Due to this fact, GB phases are sometimesreferred to as ‘complexions’52–54 to emphasize their connedheterogeneous nature in contrast to a bulk phase that is – bydenition – homogeneous.

Despite the enormous knowledge accumulated over the lastcentury about the fundamental aspects of GBs,39,40,55,56 itsapplication at a technical level is awaiting more comprehensivetools to allow systematic GB engineering. In particular, data-bases on GBs' properties are less popular; while thermodynamicand kinetic descriptions for most bulk phases are well estab-lished, analogous systematic and general descriptions for GBsare rarely investigated. This is because of the individualisticnature of GBs as every GB can be different depending on itscrystallographic properties.55 Although GB properties are func-tions of a large space of crystallographic variables, one may notneglect the fact that they are made of the same constituents asfor the grain interior (bulk), albeit deviating in local structure.An alternative view, therefore, can be to picture GBs withreference to their corresponding bulk phases. Thus, parallel tothe current characterization techniques57 and automatedsimulation methods58,59 that are mainly focused on studyingindividual GBs, a general approach could be assessing thethermodynamic and kinetic properties of GBs with reference tothe known bulk. For this purpose, we need to establish a phys-ical framework that allows an approximation of the GB envi-ronment with respect to its reference bulk material.

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In the current study, I propose a continuous atomic densityeld and its spatial variations to describe the GB region, relativeto its adjacent homogeneous bulk material, in a regularsubstitutional binary solution. Based on this idea, (i) the GB'sGibbs free energy is approximated, (ii) GB segregation and thecoexistence of the bulk and GB phases are discussed and, (iii)a concept for computing GB phase diagram is described. Thenovelty of the current model is that it enables a rapid, pragmaticassessment of GB properties based on available bulk thermo-dynamic data. To demonstrate the application of the currentmodel, GB segregation and interfacial phase separation in thePt–Au system are qualitatively studied. Furthermore, the rela-tionships between the relative atomic density eld and the GBnature in terms of GB excess free volume and misorientationangle are discussed. In two parallel studies, the applications ofthis model are further demonstrated in understanding segre-gation and interfacial phase separation in different alloysystems. In one case, we apply the density-based formulation torealize the Fowler–Guggenheim segregation isotherm for co-segregating Ni and Mn atoms at various types of GBs ina FeMnNiCrCo high-entropy alloy.60 A simpler version of themodel has been applied to studying Mn segregation in thebinary FeMn alloy system.61 The current density-based modeloffers a pragmatic approach to build GB thermodynamic data-bases, segregation isotherms and phase diagrams for studyingheterogeneous polycrystalline microstructures.

2. Density-based model for GBthermodynamics

In his seminal work, van der Waals proposed that an interfacecan be described by a continuous density eld43 and its spatialvariations. For a GB, that needs to accommodate for thegeometrical mismatch between two adjacent grains, an analo-gous picture can be portrayed in which the density of the GB willbe different than that of the bulk materials. This is, on the onehand, even a simpler case than the van der Waals's setup, as thebulk density on the two sides of the GB are the same. On theother hand, however, it can be more complex because of thecrystalline nature of the bulk material that may extend into theGB region. As a rst attempt, here we neglect the heterogeneityof GB density (the density variation within the GB plane due tothe crystallinity) and instead focus on the possibility ofconsidering an ‘average’ atomic density parameter and densitygradient terms to describe GB region.

In the following, we derive the density-based Gibbs free energyof a GB in unary (Section 2.1) and a substitutional binary regularsolution (Section 2.2) systems. To do so, we take a variationalapproach to obtain density-based free energy density and cast thisinto the conventional thermodynamic framework, as used inCALPHAD approaches. We obtain the Gibbs free energy func-tional as a function of the atomic density eld, concentrationeld, and their respective spatial variations. Using this free energydescription, we obtain the GB phase diagram and segregationisotherm, Section 3. The model is demonstrated on the Pt–Ausystem. Different aspects of the current model, i.e., the

This journal is © The Royal Society of Chemistry 2020

coexistence of the bulk and GB phases and the relationshipbetween the GB density, GB excess free volume, and GB misori-entation angle, are discussed in Section 4.

2.1 A GB in a pure substance

At constant temperature T and pressure p, the Gibbs free energyfunctional of a systemmade of pure substance A can be written as

GA ¼ðU

GA d~r ¼ðU

rn ðHA � TSAÞ d~r (1)

where rn(~r) is the atomic density eld equivalent to the inversemolar volume eld (Vm(~r))

�1, HA is enthalpy per unit mole withHA ¼ KA + EA + pVA (KA: kinetic energy, EA: potential energy andpVA: mechanical energy), and SA is entropy per unit mole,respectively. van der Waals has shown43 that the potentialenergy of a heterogeneous system depends on the (local) densityas well as (nonlocal) density gradients. In this study, weconsider a symmetric at GB separating two innitely largehomogeneous grains. For this system, breaking the symmetryonly normal to the GB plane, the interactions can be realizedconsidering only one spatial dimension. The potential energydensity will be

EAðxÞ ¼ EAð�NÞ þ 1

2

ðx�N

f ðrÞ dr: (2)

Here EA(�N) is the potential energy inside the homogeneousgrain at x ¼ �N and the second term describes the energystored in matter when brought from �N to a given position x.f(r) is the sum of all forces acting on point r:

f ðrÞ ¼ðN0

½zðrþ qÞ � zðr� qÞ� dq (3)

where z(r� q) is the force density at point r upon the interactionbetween the twomaterial points separated by a distance�q. Theinteraction forces between atoms depend on the interatomic

potential U(r � q) with zðr � qÞ ¼ vUðr � qÞvq

.

Although the detailed form of the interatomic potentials/forces can vary for different types of atoms, it is well-knownthat the atomistic interactions rapidly decays over thedistance between atoms. For the sake of our discussion, weconsider here a simple functional form

zðr� qÞ ¼ a nðr� qÞqz

¼ jðqÞrnðr� qÞ (4)

where a is a material constant, n(r� q) is the number of atoms, z

is a positive exponent and we dene jðqÞ ¼ aVL

qz. Eqn (4) with

z [ 1 is inspired by interatomic force relations in the absenceof the long-range electrostatic interactions.62 Here VL is thecharacteristic coarse-graining volume over which the mesoscale

atomic density eld is measured by rnðrÞ ¼nðrÞVL

. Similar coarse-

graining descriptions are worked out in dening mesoscale freeenergy formulations.63,64 The specic form of j(q) can be obtainedfrom atomistic simulations. The second- or higher-order depen-dencies of the interaction forces on the atomic density (including

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repulsive atomic interactions) can also be assumed. In the currenttreatment, however, we limit ourselves to the simplest rst-orderlinear approximation as in eqn (4).

In the theories of atomistic simulations, the atomic forcesare practically calculated up to a cut-off radius Rc beyond whichthe interatomic forces are neglected.62 Hence, one reasonableapproximation for the coarse-graining volume would be

VL z4p3Rc

3 which can relate the atomistic simulation length-

scale with the current mesoscale density-based formulation.Inserting eqn (4) in eqn (3), and integrating by part we can write

f ðrÞ ¼ðN0

jðqÞ ½rnðrþ qÞ � rnðr� qÞ� dq

¼ jðqÞ ½rnðrþ qÞ � rnðr� qÞ�N0

�ðN0

jðqÞ v½rnðrþ qÞ � rnðr� qÞ�vq

dq

¼ �ðN0

jðqÞ v½rnðrþ qÞ � rnðr� qÞ�vq

dq (5)

with limq/N

jðqÞ ¼ 0 and limq/0

½rnðr þ qÞ �rnðr� qÞ�¼ 0. Applying

the Taylor expansions rnðr � qÞ ¼ rnðrÞ �vrn

vrqþ 1

2!v2rn

vr2q2

� 13!

v3rn

vr3q3 þ 1

4!v4rn

vr4q4 � 1

5!v5rn

vr5q5 þ., eqn (5) gives

f ðrÞ ¼ �ðN0

2 jðqÞ�vrn

vrþ 1

2!

v3rn

vr3q2 þ 1

4!

v5rn

vr5q4 þ.

�dq

¼ 2E0A

�vrn

vr

�� kA

�v3rn

vr3

�� k

0A

�v5rn

vr5

��.

(6)

where we dene E0A ¼ �

ðN0jðqÞ dq , kA ¼

ðN0jðqÞq2 dq and

k0A ¼ 1

12

ðN0jðqÞq4 dq . Using eqn (2) and (6) with the boundary

values EA(�N) ¼ E0Arn(�N),�v2rn

vr2

��N

¼ 0 and�v4rn

vr4

��N

¼ 0,

we obtain the potential energy as a function of atomic density:

EAðxÞ ¼ E0ArnðxÞ �

kA

2

�v2rn

vr2

�x

� k0A

2

�v4rn

vr4

�x

�. (7)

As a consequence of the Tylor expansions, it is clear that thesixth and higher-order spatial derivatives of the atomic densityeld can still contribute to the potential energy. These are,however, neglected in the following. Thus, the free energyfunctional in eqn (1) can be written as

GA ¼ðU

"E0

Arn2 þ ðKA þ pVA � TSAÞ rn þ

kA

2ðVrnÞ2

þ k0A

2

�V2rn

�2 #d~r (8)

where we use a three-dimensional notation,Ð(Vrn)

2 dV ¼ �Ðrn-V2rn dV and

Ð(V2rn)

2 dV ¼ �ÐrnV4rn dV with Vrn ¼ 0 and V3rn ¼0 at the boundaries of the integrals. Here two density-dependent

26730 | RSC Adv., 2020, 10, 26728–26741

terms (KA + pVA � TSA)rn and E0Arn2 appear in the free energy

functional that scale differently with respect to the density rn.The bulk atomic densities on the two sides of the GB can

have different isotropy, depending on the crystallographicplanes that meet at the GB plane. This can result in asymmetricGB structure and density prole. In the current study, however,we neglect this possibility and assume the same bulk density onthe two sides of the GB. Thus we can further simplify ourdescription by dening a dimensionless density parameter

rðxÞ ¼ rnðxÞrBn

with rBn ¼ rn(�N) and writing

GA ¼ EBAr

2 þ �KB

A þ pVBA � TSB

A

�rþ kr

2ðVrÞ2 þ k

0r

2

�V2r�2(9)

in which EBA ¼ E0A(rBn)

2, kr ¼ kA(rBn)

2, k0r ¼ k

0A ðrBnÞ2, KB

A ¼KAr

Bn, pV

BA ¼ pVAr

Bn, S

BA ¼ SAr

Bn and, the Gibbs free energy of the

homogeneous bulk phase (for r ¼ 1) recovers as

GBA ¼ EB

A + KBA + pVB

A � TSBA ¼ HB

A � TSBA. (10)

To obtain the GB free energy, one can compare a heteroge-neous system including a GB against a homogeneous bulksystem (without the boundary). Thus, subtracting eqn (10) fromeqn (9) we obtain

GGBA ¼ EB

A

�r2 � 1

�þ �KBA þ pVB

A � TSBA

�ðr� 1Þ

þkr

2ðVrÞ2 þ k

0r

2

�V2r�2: (11)

Eqn (11) gives an approximation of the GB Gibbs free energydensity, GGB

A . For the symmetric at GB centered at position x ¼0, the equilibrium density prole across the GB region (for 0# x# h) will be

rðxÞ ¼ rGB �X4i¼1

Ci þ C1eb1x þ C2e

�b1x þ C3eb2x þ C4e

�b2x (12)

which satisesdGGB

A

dr¼ 0 with k

0r . 0. The coefficients Cis need to

fulll boundary conditions r(x ¼ 0) ¼ rGB, r(x ¼ h) ¼ 1 and the

continuity conditionsvr

vxðx ¼ 0Þ ¼ vr

vxðx ¼ hÞ ¼ 0. Here

b1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr2 � 8EB

Ak0r

q2k0

r

vuut, b2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr2 � 8EB

Ak0r

q2k0

r

vuutand h is

the GB half width. For a simpler case, assuming k0r ¼ 0, the

equilibrium density prole across the GB region reads

rðxÞ ¼�1þ rGB

2

���1� rGB

2

�cos

�px

h

�(13)

with h ¼ p

ffiffiffiffiffiffiffiffiffiffiffikA

�2EBA

r. Eqn (12) or (13) give a continuous atomic

density prole across the GB. For more details on theseequations see Appendix A. The continuity of atomic densityprole across the GB region is conrmed by atomistic simula-tions.61 Fig. 1 compares the atomic density prole eqn (13)

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versus previous phase-eld models for GBs47–50 (Fig. 1(c)) as wellas the classical phase-eld prole across an interface65–67

(Fig. 1(d)). Using eqn (13) with k0r ¼ 0, the GB energy

g ¼ 2ðh0GGBA dr can be analytically obtained as

g ¼ a0(1 � rGB)2 (14)

with a0 ¼ p

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2EB

AkA

q. This equation approximates the direct

relation between g and rGB for an equilibrium density prole

Fig. 1 (a) and (b) Continuous density profile across a symmetric flat GBwith a density r(x ¼ 0) ¼ rGB (eqn (13)). The GB width is related to themodel parameters EBA and kA. The current density profile across GB iscompared versus schematic drawing of order parameters from (c) theKWC phase-field model for GBs47 usually applied for studying GBphase transitions48–50 and (d) the classical phase-field profile (see forinstance ref. 65). (e) The energy of the GB as a function of its density(eqn (14)) is shown (see Section 4.2 for discussions).

This journal is © The Royal Society of Chemistry 2020

across the GB. Fig. 1(e) shows the normalized valueg

a0as

a function of rGB. In the following, rGB, the average relativeatomic density within the GB plane, will be referred to as ‘GBdensity’. The signicance of the GB density and its relationshipwith the GB character is discussed in Section 4.2.

2.2 A GB in a regular binary alloy

In order to extend the current density-based model for a GB ina binary alloy, we need to discuss the signicance of mixing energyas a function of atomic density. For a bulk regular solutionmade ofA (solvent) and B (solute) atoms, the change in the Gibbs freeenergy due to the mixing can be approximated as68

DGBmix(XB) ¼ DHB

mix � TDSBmix ¼ UXAXB

+ RT[XA ln XA + XB ln XB] (15)

with

U ¼ NaZD3 (16)

in which R is the gas constant,U is themixing enthalpy coefficient,Na is Avogadro number and Z is the coordination number.

D3 ¼3AB � 3AA þ 3BB

2

is bonding energy in which 3ij is the

bonding energy between atoms i and j and XA (XB) is the molefraction of atoms A (B). In principle, the mixing energy terms ineqn (15) can be assumed to follow the same scaling found in eqn(9). The effect of changing the coordination number Z on the GBenthalpy of mixing and GB segregation has been previously dis-cussed (see for instance ref. 40, 46 and 69 and references therein).While the coordination number is proportional to the localdensity, the stretch in the bonding energy D3 is also expected tochangewith the density in the limit of linear elasticity. Consideringthese facts and neglecting the dependency of congurationalentropy for the sake of simplicity, we can write

DGmix(XB, r) z r2DHBmix � TDSB

mix. (17)

Following Cahn and Hilliard,44 an energy term due to theconcentration gradients – in the presence of chemical hetero-geneity – enters the free energy description as well. Thus, usingeqn (9) and (17), the density-based Gibbs free energy of a regularsolid solution (SS) can be written as

GSSðXB; rÞ ¼ XAGAðrÞ þ XBGBðrÞ þ DGmixðXB; rÞ

¼ XA

EB

Ar2 þ �KB

A þ pVBA � TSB

A

�rþ kA

2ðVrÞ2 þ k

0r

2

�V2r�2!

þXB

EB

Br2 þ �KB

B þ pVBB � TSB

B

�rþ kB

2ðVrÞ2 þ k

0r

2

�V2r�2!

þ r2UXAXB � TDSBmix þ

kX

2ðVXBÞ2

(18)

where kX is the concentration gradient coefficient. In eqn (18),correspondingmolar fractions XB

i and XGBi will be considered for

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the bulk and GB regions, respectively. For the homogeneousbulk phase, one recovers

GBSS(X

BB, r ¼ 1) ¼ XB

A(HBA � TSB

A) + XBB(H

BB � TSB

B)

+ UXBAX

BB � TDSB

mix. (19)

Using eqn (18) we are able now to study the GB phaseequilibria and segregation isotherms as presented in the nextsection.

3. Applications of the density-basedmodel3.1 Equilibrium GB phase diagram

Obviously, a GB may not exist without its corresponding bulkphase(s). It is, however, useful to study the equilibrium phasediagram of a hypothetical GB as a function of its atomic density.As eqn (18) suggests, for such hypothetical GB with r < 1 thephase equilibria should differ from that of the correspondingbulk material with r ¼ 1. The equilibrium states are evaluatedby minimizing the Gibbs free energy functional with respect to

Fig. 2 Gibbs free energy density of the Pt–Au system (eqn (22)) isshown as a generalized function of composition and atomic density atT¼ 300 K. For rGB¼ 1¼ rB, the Gibbs free energy of the reference bulkis recovered. For rGB < 1 the free energy increases.

Fig. 3 The Gibbs free energy density of the bulk and our hypotheticalcompared at different temperatures. As expected, the GB Gibbs free en

26732 | RSC Adv., 2020, 10, 26728–26741

the concentration and atomic density elds, i.e.dGalloy

dr/0 and

dGalloy

dXB/0 with

dGalloy

dq¼ vGalloy

vq� V

vGalloy

vVqþ V2 vGalloy

vV2qand

Galloy ¼ÐGSS d~r. A numerical solution of these equations can

provide information about the spatiotemporal evolution ofphases in the GB region. This has been quantitatively demon-strated in a parallel study on the Mn segregation in Fe–Mnsystem.61 The aim of the current study, however, is to provideanalytical descriptions for equilibrium GB thermodynamics.Hence, in the following, we rather investigate a single materialpoint at the GB center where spatial gradients of the atomicdensity eld vanish, i.e. Vr(x ¼ 0) ¼ 0. We further assume thatfor this point the concentration gradient and higher-orderdensity gradients can be neglected. Thus, with r(x ¼ 0) ¼ rGB

and XB(x ¼ 0) ¼ XGBB , the density-based Gibbs free energy

description at this material point simplies to

GSS(XGBB , rGB) ¼ XGB

A (EBA(r

GB)2 + (KBA + pVB

A � TSBA)r

GB)

+ XGBB (EB

B(rGB)2 + (KB

B + pVBB � TSB

B)rGB)

+ (rGB)2UXGBA XGB

B � TDSBmix. (20)

Eqn (20) allows to approximate the GB thermodynamicproperties, analytically. To demonstrate application of thecurrent density-based model, we consider a hypothetical GBwith rGB ¼ 0.75 in the Pt–Au system. In this alloy system, theonly difference to the theoretical regular solution is in theenthalpy of mixing that extends to the second term in Redlich–Kister polynomial:

UPtAu ¼ L0 + L1(XBPt � XB

Au). (21)

Inserting eqn (21) in eqn (20) we obtain

GSS(XGBAu , r

GB) ¼ XGBPt (E

BPt(r

GB)2 + (KBPt + pVB

Pt � TSBPt)r

GB)

+ XGBAu (E

BAu(r

GB)2 + (KBAu + pVB

Au � TSBAu)r

GB)

+ (rGB)2UPtAuXGBPt X

GBAu � TDSB

mix. (22)

Grolier et al.70 have assessed the bulk thermodynamic datafor binary the Pt–Au and reported L0 ¼ 11 625 + 8.3104T and L1¼ �12 616 + 5.8186T [J mol�1] for the FCC Pt–Au solid solution.The rest of the thermodynamic data for pure Pt and Au are

GB with rGB ¼ 0.75 in the Pt–Au system obtained from eqn (22) areergy density is higher than the bulk.

This journal is © The Royal Society of Chemistry 2020

Fig. 4 Equilibrium phase diagram for the bulk r ¼ rB ¼ 1 and ourhypothetical GB with r ¼ rGB ¼ 0.75 are obtained for the Pt–Ausystem. The dash lines represent the chemical spinodals. The coloredarea marks the possible domain where GB phase diagrams for 0.75 <rGB < 1 can appear. Note that the equilibrium bulk and GB phasediagrams are plotted independently. The coexistence of the bulk andGB phases will be discussed in Sections 3.2 and 4.1 and Fig. 6.

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extracted from SGTE compilation by Dinsdale.71 Inserting thesevalues in eqn (21) and (22), one obtains the Gibbs free energy asa function of GB density and composition. Fig. 2 shows a 3D

XGBB

1� XGBB

¼ XBB

1� XBB

� exp

� ½DEB þ U�ðrGB2 � 1Þ þ ðDKB þ pDVB � TDSBÞðrGB � 1Þ þ 2U

�XB

B � rGB2XGB

B

�RT

!: (23)

map of the Gibbs free energy for the Pt–Au system at 300 K asa function of composition XAu and GB density rGB. We foundthat as the GB density decreases, the Gibbs free energy increasesand deviates larger from the corresponding bulk values. Fig. 3compares bulk and GB Gibbs free energy for rGB ¼ 0.75 atdifferent temperatures.

XGBAu

1� XGBAu

¼ XBAu

1� XBAu

� exp

�� ½DEB þ L0�ðrGB2 � 1Þ þ ðDKB þ pDVB � TDSBÞðrGB � 1Þ þ 2L0

�XB

Au � rGB2XGB

Au

�þQ

RT

�(24)

Using eqn (22), the equilibrium phase diagrams of thebulk and any specic standalone GB (with a given GB density)in the Pt–Au system can be generated. Fig. 4 compares thephase diagrams of the bulk and our hypothetical GB (with rGB

¼ 0.75) in the Pt–Au solid solution. The results show that,inside a (hypothetically standalone) GB, the interfacial phaseseparation becomes possible but for smaller ranges oftemperature and composition when compared to the bulkphase diagram.

As it is evident from the Gibbs free energy plots (Fig. 2 and3), the GB thermodynamics strongly depend on the GB density

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rGB. Although the phase diagram is a nonlinear function of theGB density, the possible values for GB density rGB are niteand vary close to 1. For instance, the colored area in Fig. 4indicates where the GB phase diagram can appear for 0.75 <rGB < 1. Thus, once the range of GB density is determined, theequilibrium GB phase diagrams can be approximated. The GBdensity represents the nature (type) of the GB and correlateswith the GB energy and misorientation angle. These aspectsof the current density-based model are discussed inSection 4.2.

3.2 GB segregation behavior

In Section 3.1 a hypothetical standalone GB was discussed. Fora complete description of GB phase equilibria, however, thecoexistence between the GB and bulk phases must be under-stood. In a binary system, this requires equality of the (relative)chemical potentials mBB � mBA ¼ mGBB � mGBA all across the system(parallel tangent construction).24,72 In reality, these are notsufficient conditions as a minimum energy state with respect tothe atomic density eld must be satised as well. However, if weconsider the same material point at the GB center as above, wecan simplify the problem and solve for the equality of the(relative) chemical potentials using eqn (20) which gives

In eqn (23), XGBB is the composition of the GB with r ¼ rGB, XB

B isthe composition of the bulk far from the GB, DEB¼ EBB� EBA, DK

B

¼ KBB � KB

A, pDVB ¼ pVBB � pVBA, and DS ¼ SBB � SBA. Eqn (23)

resembles the Fowler–Guggenheim segregation isotherm24 butalso takes the specic effect of a given GB, represented by itsdensity rGB, into account. For the Pt–Au system, we obtain

with Q ¼ L1½2rGB2XGBAu ð1� XGB

Au Þ � ð1� 2XGBAu Þ2 � 2XB

Auð1� XBAuÞ

þð1� 2XBAuÞ2�, DEB ¼ EBAu � EBPt, DK

B ¼ KBAu � KB

Pt, pDVB ¼ pVBAu �

pVBPt, and DS ¼ SBAu � SBPt. Eqn (23) and (24) use GB density andbulk thermodynamic data as the inputs and give the corre-sponding GB segregation isotherms.

Fig. 5 presents the (relative) chemical potentials and thesegregation isotherms for the bulk and the exemplar GB withrGB ¼ 0.75 in the Pt–Au system at three different temperatures.The chemical potential of the GB which is a function of GBdensity differs from the corresponding bulk chemical poten-tial. The Maxwell construction (dening the two-phase

RSC Adv., 2020, 10, 26728–26741 | 26733

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regions) and the spinodal compositions for both GB and bulkmaterials are built. The graphs show that the GB spinodaldecomposition (interfacial spinodal) can occur for lowerchemical potential values below the bulk spinodal (e.g. inFig. 5(a)). As a result, the two-phase GB can be in equilibriumwith a single-phase bulk. As a matter of fact, it has been re-ported that a segregation-assisted interfacial phase separationcan occur in Pt–Au system which results in the formation oftwo-phase GBs coexisting with a single-phase bulk.73,74 Theresults in Fig. 5 suggest that the two-phase bulk will be inequilibrium with a single-phase GB. In fact, at a giventemperature, for any bulk composition within the two-phaseregion of the phase diagram, the two-phase bulk is expectedto be in equilibrium with a single phase/composition GB.These are because of a vertical shi in the GB chemicalpotential (with respect to the bulk chemical potential) that canbe seen in Fig. 5(a). Combining these results with the equi-librium bulk and GB phase diagrams (Section 3.1), we are ablenow to study the complete GB thermodynamic phase diagram.This is discussed in Section 4.1.

4. Discussion4.1 Coexistence of GB and bulk phases

In this model, the GB region is described by a continuousatomic density and its spatial gradients. For the sake ofsimplicity, here we have limited our derivations to the rst-order approximation of the potential energy with respect to r

Fig. 5 (a)–(c) The chemical potentials of the Au atoms in bulk and a GB (w(f) Using the parallel tangent condition the segregation isotherms aresegregation appear depending on the GB density and temperature. See

26734 | RSC Adv., 2020, 10, 26728–26741

and its spatial variations. The current density-based modelprovides a simple method for approximating the Gibbs freeenergy of a given GB based on available bulk thermodynamicdata. The continuity of the atomic density eld across GB allowsfurther simplifying the Gibbs free energy description at the GBplane where the gradient term Vr vanishes. The results of thecurrent study are obtained for this single material point withinthe GB region. The model can be applied in full-eld simula-tions of GBs as well, to evaluate the temporal evolution of theatomic density and concentration elds. In two parallel study,we have demonstrated the application of the density-basedmodel for studying GB segregation in a FeMnNiCrCo highentropy alloy60 and segregation engineering of Fe–Mn alloys.61

In the current study, we focus on the equilibrium GB prop-erties in a binary substitutional solid solution. Using thedensity-based formulation we extend the concept of the phasediagram for GBs, Fig. 6(b). To arrive there, we started with theequilibrium phase diagram for the bulk and a hypotheticalstandalone GB, with rGB ¼ 0.75, in the Pt–Au solid solution(Fig. 4). The results show that similar to the bulk materials, GBsin the Pt–Au system can also undergo an interfacial spinodalphase separation. In this case, we show that GB two-phaseregion is conned to smaller temperature and compositionranges. As the GB density deviates larger from the bulk density,the GB miscibility gap in the phase diagram becomes smaller.

The coexistence of the bulk andGB phases can be studied usingthe equal chemical potential condition.24,72 Here the bulk phase,which is the dominant part of the microstructure, is assumed to

ith rGB ¼ 0.75) Pt–Au are plotted for three different temperatures. (d)–obtained for the corresponding temperatures. Different regimes ofdiscussions in Section 4.2.

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Fig. 6 (a) For Pt–Au system at 700 K, a jump (GB spinodal) occurs before the bulk spinodal decomposition takes place. This is a zoomed in plot inFig. 5(d) (left corner), see also Fig. 5(a). (b) Based on the current density-basedmodel the coexistence of the bulk and GB phases is depicted in thephase diagram for the Pt–Au system. A two-phase bulk will be in equilibriumwith a single-phase GBwhile a two-phase (spinodally-decomposed)GB is shown to be in equilibrium with a single-phase bulk.

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dedicate the ultimate thermodynamic states and behavior of theGBs. According to eqn (23) (or eqn (24) for the Pt–Au system), theGB segregation level depends on the temperature, bulk composi-tion, as well as GB type represented by its density rGB. For a regularsolution discussed here, the driving forces for the segregation canbe divided into two parts: the rst contribution is the ideal segre-gation energy DEBðrGB2 � 1Þ þ ðDKB þ pDVB � TDSBÞðrGB � 1Þthat is analogous to the segregation driving force in Langmuir–McLean isotherm.72 If we neglect the energetic contributions dueto the mixing (U ¼ 0), we obtain a density-based version of Lang-muir–McLean relation as:

XGBB

1� XGBB

¼ XBB

1� XBB

�exp

�� DEBðrGB2 � 1Þ þ ðDKB þ pDVB � TDSBÞðrGB � 1Þ

RT

�(25)

that describes solute segregation for a given GB in an idealsolution. The second contribution in eqn (23) isUðrGB2 � 1Þ þ 2U½XB

B � rGB2XGB

B � due to the mixing whichcontains a term with the GB concentration XGB

B . The twocontributions discussed above can have cooperative orcompetitive effects on the segregation level, depending on themagnitude and sign of U and DEB.

Fig. 5 presents the chemical potentials and segregationisotherms for the Pt–Au system. In this system, three regimes ofbulk-GB coexistence could be identied as a function of the bulkcomposition. First, segregation to the GB occurs as the Au contentincreases in Pt-rich alloys (regime I). For a two-phase bulk, the GBcomposition becomes xed (regime II) that applies to the entirerange of bulk composition in the two-phase bulk region. Anyfurther increase of the Au content results in solute depletion in theGB (regime III) with respect to the bulk. In Fig. 5(a) and (d)different regimes of segregation are marked.

This journal is © The Royal Society of Chemistry 2020

For a Pt-rich Pt–Au alloy (regime I), the GB, which is enriched inAu due to the segregation, can undergo an interfacial spinodaldecomposition before a bulk spinodal becomes possible. At 700 K,for instance, a jump in the GB segregation, associated with theinterfacial phase separation, is revealed in the Pt–Au system.Fig. 6(a) shows this jump in the GB composition as a function ofbulk composition (this is a zoomed-in plot of the le corner inFig. 5(d)). At this jump, the GB decomposes into low-concentrationand high-concertation domains that occurs in the presence ofa single-phase bulk. This means that for a specic bulk composi-tion (in the single-phase region of the bulk phase diagram) a spi-nodally-decomposed two-phase GB will form. Consistent withthe current predictions, a segregation-assisted interfacial spinodaldecomposition has been indeed reported in nanocrystalline Pt–Aualloy.73 Also, atomistic simulations of the Pt–Au system conrmedsuch interfacial spinodal phase separation for bulk compositionswell below the bulk spinodal range.74 GB spinodal phase separa-tion is also evidenced in other material systems with technologicalsignicance.75,76

For the two-phase Pt–Au bulk materials (regime II), themodel predicts that a single-phase GB comes to coexist. This isbecause of the vertical shi in the GB chemical potential withrespect to the corresponding bulk chemical potential curve, asdepicted in Fig. 5(a). In practice, a range of atomic densities andcompositions exists in the GB region (between the GB plane andbulk interior) that can enable interfacial phase separation in theGB region. Combining the equilibrium GB phase diagram(Fig. (4)) and the GB segregation isotherms (Fig. (5)), we canobtain the complete GB phase diagram as presented in Fig. 6(b).

The discontinuous jump in the GB segregation isotherm(Fig. 6(a)) is a result of the difference in the free energy functionaland chemical potentials of the bulk and GB phases, captured bythe density-based model. Depending on the sign of DEB and theshape of U(XB) function (in composition space), the GB chemicalpotential and its corresponding spinodal decomposition can lie

RSC Adv., 2020, 10, 26728–26741 | 26735

Fig. 7 The GB chemical potential can be ‘Below’ or ‘Above’ the bulk'scurve. XBS1B and XBS2B show the bulk equilibrium compositions in thetwo-phase region. The two-phase GB will be in equilibrium witha single-phase bulk material as marked by the arrows. The dots on thebulk chemical potential curve indicates the single-phase GB is equi-librium with the two-phase bulk.

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‘Above’ or ‘Below’ the bulk spinodal (Fig. 7). In either case, one canactivate a segregation-assisted interfacial phase separation, thatcan act as a precursor for subsequent nucleation, before reachingthe bulk two-phase region. It has been shown that a segregation-assisted transient spinodal phase separation comes to exist atGBs for a larger range of bulk compositions61 which enablesmanipulation of GB segregation by desired heat treatments.

In a polycrystalline material having a large population of GBsof various types (with different crystallographic properties), theGB densities are expected to differ from one GB to another. Aquantitative application of the current model to technicalmaterials, therefore, requires determination of the GB structureand densities for different types of GBs. For this purpose,atomistic simulations, for instance, can provide the necessarymeans to evaluate GB density proles. Though atomisticsimulation of GBs is beyond the scope of this study, in thefollowing we briey analyze the relationships between the rGB,GB excess free volume and GB misorientation angle that canprovide more insights to the density-based concept.

4.2 The relation between the GB density and GB nature

In order to make use of the current model, it is helpful toexplore the signicance of the average GB density and its rela-tionship with the GB nature. Using atomistic simulations andan appropriate numerical coarse-graining scheme, 3D densitymap of any GB can be extracted from its equilibrium atomicconguration that gives h and rGB.61 GB energy g can be relatedto the h and rGB values. Eqn (14) approximates the relationshipbetween GB energy and its density rGB when k

0r ¼ 0 is assumed.

It is observed that GBs with higher energies, i.e., lower GBdensities, attract more solute content during the segregation.77

Using eqn (14) and (20), the GB Gibbs free energy of a regularsolution can be written as a function of its initial GB energy.Accordingly, we can rewrite eqn (23) as

XGBB

1� XGBB

¼ XBB

1� XBB

�exp

0BB@�

½DEB þ U���

1�ffiffiffiffiffig

a0

r �2� 1

�þ ðDKB þ pDVB � TDSBÞ

��1�

ffiffiffiffiffig

a0

r �� 1

�þ 2U

�XB

B ��1�

ffiffiffiffiffig

a0

r �2XGB

B

�RT

1CCA (26)

that gives a useful approximation for describing GB's tendencyfor segregation as a function of its initial energy.

One way to elaborate about the GB density rGB is to nd itsrelationship with the GB excess free volume dened as78

DV ¼ WNB

N0

�1� NGB

NB

�: (27)

Here W is the atom diameter, NGB is the number of atoms in theGB,NB is the number of atoms in the bulk within the same volumeand N0 is the number of atoms per unit area of the GB. In prin-ciple, the ‘excess’ volume represents the ‘shortage’ of atoms in theGB plane. On the other hand, comparing a GB against the bulk of

26736 | RSC Adv., 2020, 10, 26728–26741

the same volume and using eqn (13), the relative number of atoms

in a GB will be nGB ¼ ð1þ rGBÞ2

(where nB ¼ 1) and the relative

difference (with respect to the bulk) will be

Dn ¼ 1 � rGB (28)

where n ¼ 2ðh0r dr is the (relative) total number of atoms in the

GB. Comparing eqn (27) and (28) one can nd the one-to-onerelationship between Dn and DV, where we can identify DV f (1� rGB). Extensive works have been devoted to calculate andmeasure GB excess free volume (see for instance ref. 79–81). Aaronand Bolling78 have shown that the energy of different types of GBscan be associated with their excess free volume which can beextended as well for the GB density parameter rGB.

The lower limit of the GB density value (highest excess freevolume) shall be obtained for general high angle GBs (HAGBs).Aaron and Bolling remarked that the excess free volume ina general HAGB is comparable to that of a liquid phase.78 On theother hand, the maximum GB density value is the correspond-ing bulk density rB ¼ 1. Special GBs such as coherent twinboundaries (TBs) with low coincidence values show small GBenergy and excess free volumes.78 This is well consistent withthe current model in which as DV goes to zero, rGB approachesrB ¼ 1 and g approaches zero (eqn (14)).

For a low-angle GB (LAGB), the situation becomes a bit morecomplex because of the dislocations. Since dislocations are

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localized defects with a lower atomic density in their core, oneexpects the local GB density rGB within the LAGB plane to uctuateaccordingly. In addition, elastic interactions between dislocationscan play a role in determining rGB for LAGBs. For a simple caseneglecting these elastic interactions, we can approximate theaverage GB density for a LAGB by a simple volume averaging ofregularly spaced dislocations, as detailed in Appendix B. For a tiltGB, a simple geometrical construction of a LAGB gives the average

GB density rGBtilt z 1� sin q

4with the GB misorientation q, and the

edge dislocation core density rD z 0.75. For a larger misorienta-tion angle, the GB density will be smaller, deviating larger frombulk properties. We also obtain a misorientation-dependent

Dnzsin q

4and thus a corresponding GB excess free volume DV

f sin q that is consistent with the previous studies.78,82 GB segre-gation as a function of GB misorientation has been studied in thePt–Au system. Seki et al.83,84 and Seidman85 studied Au segregationto twist boundaries and have shown that although the GB segre-gation is not homogeneous, its average level increases as themisorientation angle increases (with the exception of special GBs).These observations conrm the results from the current density-based model where a higher misorientation angle correspondsto a lower GB density. An in-depth study of LAGBs requiresconsideration of elastic energy contributions which is le fora future study.

5. Summary and outlook

In this work, we have derived a density-based model for assessingGB thermodynamics. The current model uses available bulkthermodynamic data, as input, and gives a rather general formu-lation for GB's Gibbs free energy. GB segregation isotherm andphase diagram for a regular binary solid solution have been ob-tained. We further discussed the relationships between GBdensity, GB excess free volume and, GB misorientation angle. Theresults are qualitatively demonstrated on the Pt–Au system. Weshow that in a Pt-rich Pt–Au alloy, Au atoms tend to segregate tothe GB. The average level of segregation increases with decreasingGB density, i.e., increasing GB misorientation angle (with theexception of special GBs). We also show that an interfacial spino-dal decomposition can occur in these alloys for single-phase bulkalloys. A comparison with previous studies on the Pt–Au systemconrms these predictions.

The current density-based model offers a simple, pragmaticapproach to develop GB thermodynamic databases. Despite itssimplicity, applications of this model can unroll new microstruc-ture design concepts by segregation engineering of GBs.60,61 Asystematic atomistic study of the coarse-grained atomic densityproles for different types of GBs can enrich the development ofthis model. In order to assess engineering alloys with interstitialsolute atoms, the density-based free energy description needs to beextended for multi-component materials and including elasticenergy contributions.

This journal is © The Royal Society of Chemistry 2020

AppendixA Equilibrium GB density prole

The equilibrium GB density prole across a symmetric at GBin a pure substance can be obtained by minimizing the GB freeenergy functional with respect to the relative atomic density. Fora 1D setup,

dGA

dr¼ vGA

vr� d

dx

vGA

v

�vr

vx

�0BB@

1CCAþ d2

d2x

vGA

v

�v2r

vx2

�0BBB@

1CCCA

¼ 2EBArþ CB

A � krv2r

vx2þ k

0r

v4r

vx4¼ 0 (A.1)

where CBA ¼ KB

A + pVBA � TSBA (see eqn (9)). The general solution for

this higher-order linear ordinary differential equation reads

rðxÞ ¼ � CBA

2EBA

þ C1eb1x þ C2e

�b1x þ C3eb2x þ C4e

�b2x (A.2)

with b1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr2 � 8EB

Ak0r

q2k0

r

vuutand b2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikr2 � 8EB

Ak0r

q2k0

r

vuut.

We also obtain

vrðxÞvx

¼ C1b1 eb1x � C2b1e�b1x þ C3b2e

b2x � C4b2e�b2x: (A.3)

Applying the boundary conditions r(x¼ 0)¼ rGB, r(x¼ h)¼ 1

and the continuity conditionsvr

vxðx ¼ 0Þ ¼ 0 and

vr

vxðx ¼ hÞ ¼ 0,

we obtain

rGB þ CBA

2EBA

¼ C1 þ C2 þ C3 þ C4 (A.4)

1þ CBA

2EBA

¼ C1eb1h þ C2e

�b1h þ C3eb2h þ C4e

�b2h (A.5)

C1 � C2

C3 � C4

¼ �b2

b1

(A.6)

C1eb1h � C2e

�b1h

C3eb2h � C4e�b2h¼ �b2

b1

(A.7)

The solution to eqn (A.4)–(A.7) can be computed numeri-cally. The continuity of the average atomic density prole acrossthe GB region is conrmed using atomistic simulations.61 Fora simpler case assuming k

0r ¼ 0, we nd another general solu-

tion for eqn (A.1):

rðxÞ ¼ � CBA

2EBA

þD1 cos

ffiffiffiffiffiffiffiffiffiffiffiffi�2EB

A

kr

sx

!þD2 sin

ffiffiffiffiffiffiffiffiffiffiffiffi�2EB

A

kr

sx

!(A.8)

where D1 and D2 are constants and

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Fig. 8 (a) The core of an edge dislocation has a lower density than that of the bulk that is captured by a simple geometrical analysis. GB density (b)and energy (c) as a function of the misorientation angle, with the exception of special GBs, are plotted (eqn (B.4) and (B.5)).

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vr

vx¼

ffiffiffiffiffiffiffiffiffiffiffiffi�2EB

A

kr

s "�D1 sin

ffiffiffiffiffiffiffiffiffiffiffiffi�2EB

A

kr

sx

!þD2 cos

ffiffiffiffiffiffiffiffiffiffiffiffi�2EB

A

kr

sx

!#

(A.9)

Applying the boundary conditions to this solution we obtain

D1 ¼ � 1� rGB

2(A.10)

D2 ¼ 0 (A.11)

h ¼ p

ffiffiffiffiffiffiffiffiffiffiffiffikA

�2EBA

r(A.12)

CBA

EBA

¼ �1þ rGB

2(A.13)

Although solution eqn (A.8) is limiting, it provides a possi-bility for further analytical treatment of the density prole andtherefore is used for approximating GB energy, eqn (14). Formore details see Section 2.1.

B A simple geometrical calculation of GB density

The objective of this appendix is to provide a simple analysis ofthe GB density and its dependence on the misorientation angle.We consider a LAGB that can be mapped with a set of regularlyspaced dislocations separated by a distance68

D ¼ b

sin q(B.1)

in which b is the Burgers vector of the dislocations and q is theGB misorientation angle. The ‘average’ relative GB density canbe obtained by averaging the density of a dislocation in the unitarea of the LAGB. For a tilt LAGB one obtains

rGBtilt ¼ Db� bb

Dbþ bb

DbrD (B.2)

where rD is the relative atomic density at the core of an edgedislocation. Inserting eqn (B.1) in eqn (B.2)

rGBtilt ¼ 1 � (1 � rD)sin q. (B.3)

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Here it is assumed that the volume between the two dislo-cations has the same density as bulk (rB ¼ 1). For an edgedislocation depicted in Fig. 8(a), one can approximate

rD z34

�1þ b

4a� b

�where a is the interatomic distance

normal to the edge dislocation line – here we neglect the effectof elastic energy of the dislocation on its core density.Neglecting the second term inside the parenthesis then we haverD z 0.75 that gives the atomic density of a tilt GB as

rGBtilt z 1� sin q

4(B.4)

In this relation, as q approaches zero, rGB approaches rB ¼ 1and the GB energy (according to eqn (7)) decreases towards zero.Inserting eqn (B.4) into eqn (14), we obtain

gtilt ¼ a0

ðsin qÞ216

(B.5)

that is the tilt GB energy as a function of the misorientationangle. Eqn (14), (B.4) and (B.5) give the relationship between theGB density, energy and misorientation angle based on the

current simple analysis. From eqn (28) and (B.4), Dntilt zsin q

4and thus the corresponding average GB excess free volume will beproportional to the misorientation angle as DV f sin q. Fig. 8(b)and (c) show the variation of GB density and energy as a function ofthe misorientation angle according to this analysis.

Although eqn (B.4) and (B.5) are based on a very simpliedaveraging method that does not account for the elastic energyof the dislocations, the current analysis is solid in establishingthe trend in GB densities as a function of its misorientationangle, with the exception of special GBs. It is useful to showthat the GB density converges to the expected GB energies inthe limiting cases, consistent with the previous derivations ofGB energy based on the excess free volume concept.78 In orderto connect the GB segregation isotherm with its initialmisorientation angle, we can approximate by replacing theatomic density parameter using eqn (B.4). From eqn (B.4)and (20)

This journal is © The Royal Society of Chemistry 2020

GGBSS ¼ XGB

A

EB

A

�1� sin q

4

�2

þ �KBA þ pVB

A � TSBA

��1� sin q

4

�!þ XGB

B

EB

B

�1� sin q

4

�2

þ�KBB þ pVB

B � TSBB

��1� sin q

4

�!þ�1� sin q

4

�2

UXGBA XGB

B � TDSBmix (B.6)

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Thus, using eqn (B.6) and (23) the GB segregation isothermfollows

XGBB

1� XGBB

¼ XBB

1� XBB

� exp �½DEB þ U�

��1� sin q

4

�2

�1

�ðDKB þ pDVB �TDSBÞ

��1� sin q

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�RT

0BBB@

1CCCA (B.7)

Data availability

The numerical data from this study are available upon reason-able request.

Conflicts of interest

The author declares no conict of interest.

Acknowledgements

This work is being performed under the project DA 1655/2-1within the Heisenberg program by the Deutsche For-schungsgemeinscha (DFG). The author gratefully acknowl-edges nancial supports within this program.

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